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ISSN 1088-6826(e) ISSN 0002-9939(p)

Semiclassical analysis for highly degenerate potentials

Author(s): P. Álvarez-Caudevilla; J. López-Gómez
Journal: Proc. Amer. Math. Soc. 136 (2008), 665-675.
MSC (2000): Primary 35B25, 35P15, 35J10, 31C12
Posted: November 2, 2007
MathSciNet review: 2358508
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Abstract: This paper characterizes the semi-classical limit of the fundamental energy,

$\displaystyle E(h):= \sigma_1[-h^2\Delta+a(x);\Omega],$

and ground state $\psi_h$ of the Schrödinger operator $-h^2\Delta+a$ in a bounded domain $\Omega$ , in the highly degenerate case when $a\geq 0$ and $a^{-1}(0)$ consists of two components, say $\Omega_{0,1}$ and $\Omega_{0,2}$ . The main result establishes that

$\displaystyle \lim_{h\downarrow 0} \frac{E(h)}{h^2}= \min\left\{\sigma_1[-\Delta;\Omega_{0,i}], \; i=1,2\,\right\}$

and that $\psi_h$ approximates in $H_0^1(\Omega)$ the ground state of $-\Delta$ in $\Omega_{0,i}$ if

$\displaystyle \sigma_1[-\Delta;\Omega_{0,i}]< \sigma_1[-\Delta;\Omega_{0,j}],\qquad j \in \{1,2\}\setminus\{i\}.$

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Additional Information:

P. Álvarez-Caudevilla
Affiliation: Departamento de Matemáticas, Universidad Católica de Ávila, Ávila, Spain
Email: pablocaude@eresmas.com

J. López-Gómez
Affiliation: Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040-Madrid, Spain
Email: Lopez_Gomez@mat.ucm.es

DOI: 10.1090/S0002-9939-07-09076-4
PII: S 0002-9939(07)09076-4
Keywords: Fundamental energy, ground state, highly degenerate potentials, classical conjecture of B. Simon, compact Riemann manifolds.
Received by editor(s): January 19, 2007
Posted: November 2, 2007
Additional Notes: This work was partially supported by the Ministry of Education and Science of Spain under research grants REN2003--00707 and CGL2006-00524/BOS
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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